New mathematical tools help rethink risk under uncertainty

The normal distribution has long been central to probability theory, statistics, and risk management. It underpins many classical models used to describe uncertainty in complete markets. However, when markets are incomplete and risk measures themselves are uncertain, traditional approaches may no longer apply.

In a new article published in Risk Sciences, Zengjing Chen, Xinwei Feng, and Hanchao Wang discuss nonlinear central limit theorems as a theoretical framework for understanding risk under uncertainty. Rather than assuming a single probability model, these theorems allow researchers to work with families of probability measures, including settings where the mean or variance is uncertain.

The article explains that this line of research leads to nonlinear normal distributions, a new class of limiting distributions that extend the role of the classical normal distribution. These results provide tools for studying risk in situations where standard assumptions, such as a fixed probability law or independent and identically distributed variables, may not hold.

The authors also examine limit theorems in the quantum realm. In classical random walks, the central limit theorem depends on the idea that a process can be written as a sum of independent random variables. Quantum walks behave differently: uncertainty is described through wave functions, and particle positions may converge to distributions that are not classical normal distributions. The paper notes that, in three-state quantum walks, the limiting behavior can differ substantially from two-state cases, with richer distributional patterns.

Together, these discussions highlight how modern risk science may require mathematical tools beyond the classical central limit theorem. Nonlinear central limit theorems offer a way to study risk when uncertainty affects the underlying probability model itself, while quantum-walk limit theory addresses cases where independence assumptions break down.

The article also points to applications of nonlinear expectations and nonlinear central limit theory in areas such as asset pricing in incomplete markets, sequential decision-making, reinforcement-learning-related testing problems, and optimization theory.

Contact the author:

Hanchao Wang；Institute for Financial Studies, Shandong University, Jinan 250100, China； wanghanchao@sdu.edu.cn

Conflict of interest:

The authors declare that there are no financial or personal relationships that could inappropriately influence their work, or that there are no conflicts of interest to disclose.

See the article:

Chen, Z., Feng, X., & Wang, H. Risk, Uncertainty and Nonlinear Central Limit Theorems. Risk Sciences. DOI: https://doi.org/10.1016/j.risk.2026.100055